30.16 EXERCISES
30.11 A boy is selected at random from amongst the children belonging to families with
nchildren. It is known that he has at least two sisters. Show that the probability
that he hask−1brothersis
(n−1)!
(2n−^1 −n)(k−1)!(n−k)!
,
for 1≤k≤n−2 and zero for other values ofk. Assume that boys and girls are
equally likely.
30.12 VillagesA,B,CandDare connected by overhead telephone lines joiningAB,
AC,BC,BDandCD. As a result of severe gales, there is a probabilityp(the
same for each link) that any particular link is broken.
(a) Show that the probability that a call can be made fromAtoBis1 −p^2 − 2 p^3 +3p^4 −p^5.(b) Show that the probability that a call can be made fromDtoAis1 − 2 p^2 − 2 p^3 +5p^4 − 2 p^5.30.13 A set of 2N+ 1 rods consists of one of each integer length 1, 2 ,..., 2 N, 2 N+1.
Three, of lengthsa,bandc, are selected, of whichais the longest. By considering
the possible values ofbandc, determine the number of ways in which a non-
degenerate triangle (i.e. one of non-zero area) can be formed (i) ifais even,
and (ii) ifais odd. Combine these results appropriately to determine the total
number of non-degenerate triangles that can be formed with the 2N+ 1 rods,
and hence show that the probability that such a triangle can be formed from a
random selection (without replacement) of three rods is
(N−1)(4N+1)
2(4N^2 −1)
.
30.14 A certain marksman never misses his target, which consists of a disc of unit radius
with centreO. The probability that any given shot will hit the target within a
distancetofOist^2 ,for0≤t≤1. The marksman firesnindependendent shots
at the target, and the random variableYis the radius of the smallest circle with
centreOthat encloses all the shots. Determine the PDF forYand hence find
the expected area of the circle.
The shot that is furthest fromOis now rejected and the corresponding circle
determined for the remainingn−1 shots. Show that its expected area is
n− 1
n+1
π.30.15 The duration (in minutes) of a telephone call made from a public call-box is a
random variableT. The probability density function ofTis
f(t)=
0 t< 0 ,
1
2 0 ≤t<^1 ,
ke−^2 t t≥ 1 ,wherekis a constant. To pay for the call, 20 pence has to be inserted at the
beginning, and a further 20 pence after each subsequent half-minute. Determine
by how much the average cost of a call exceeds the cost of a call of average
length charged at 40 pence per minute.
30.16 Kittens from different litters do not get on with each other, and fighting breaks
out whenever two kittens from different litters are present together. A cage
initially containsxkittens from one litter andyfrom another. To quell the